Taxonomy rewrite question

I have custom post types for dining, lodging, and activities. Each of those custom post types uses a custom taxonomy called region. If, for example, I have a region called “downtown”, I would like to be able to show dining option for that region by going to domain.com/downtown/dining. Or show hotels in that region by going to domain.com/downtown/lodging. So, basically, I like to create a rewrite rule to give the structure domain.com/tax-term/cpt-slug to show all the posts for the cpt that have that tax term selected. Any way to do this?